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Geometric approach to capture analysis of PN guidance law
Aerospace Science and Technology 12 (2008) 177–183 www.elsevier.com/locate/aescte
Geometric approach to capture analysis of PN guidance law Chao-Yong Li ∗ , Wu-Xing Jing Department of Aerospace Engineering, Harbin Institute of Technology, P.O. Box 333, Harbin 150001, P.R. China Received 21 June 2006; received in revised form 16 April 2007; accepted 17 April 2007 Available online 16 May 2007
Abstract A novel approach to capture analysis of the proportional navigation (PN) guidance law is presented in this paper. The capturabilities of the pure PN guidance law and the true PN guidance law are qualitatively studied using the differential geometric formulations. A unified postlaunch capture condition for PN guidance law is developed and expressed in terms of geometric angle and relative motion parameters instead of the initial conditions. Simulation results indicate that the proposed capture condition is more crucial at the end of the engagement. Also, the acceleration/deceleration of the missile has a distinct influence on the interception performance of the PN guidance law, and a constant thrust works viable and effectively in improving the performance of the PN guidance law. © 2007 Elsevier Masson SAS. All rights reserved. Keywords: Missile guidance; Differential geometry; Proportional navigation; Capture condition
1. Introduction Because of its inherent simplicity and ease of implementation, PN guidance law and its variants are the most widely known and used guidance schemes in the practical missile interception engagements. In attempting to solve the PN equations, two principal PN guidance laws are defined due to the directions of their accelerations. One set of definitions leads to a class of laws consisting of PPN guidance law and its variants, whose commanded accelerations are referenced relative to the missile’s velocity,1 the other class is TPN guidance law and its variants, whose commanded accelerations are referenced relative to the LOS. Moreover, it is well know that notwithstanding the relative difficulties in solving the PPN guidance problems, PPN guidance law is the most “natural” guidance law, this conclusion is drawn based on the facts that the velocity-referenced guidance laws are reasonable in practical implementation [1–3]. Moreover, in view of the importance of the PN guidance law in the missile guidance and space applications, considerable studies have been devoted to its capturability analysis * Corresponding author,
E-mail address: [email protected] (C.-Y. Li). 1 Throughout this paper, the word velocity will only be used to designate a
vector quantity; the corresponding scalar will be denoted as speed. 1270-9638/$ – see front matter © 2007 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ast.2007.04.007
in various scenarios. Shukla [3] proposed a comparative study between the TPN and PPN guidance laws, as well as their capture conditions. Yang and Chern [4] examined the relation between the navigation constant and the capture condition of the TPN guidance law. Ghose [5] developed a necessary and sufficient capture condition for TPN guidance law in the case of intercepting an intelligently maneuvering target. Song [6] and Ha [7] introduced a Lyapunov-like approach to the capturability analysis of 3D PN guidance law. Ghawghawe [8] studied the capturability of the PPN guidance law against a target executing bounded piecewise continuous time-varying maneuvers. Yang [9] and Yang [10] analyzed the influence of the navigation constant, target’s maneuver, and the initial conditions on the capture condition of 3D PN guidance law. Dimirovski [11] investigated the influence of the application of the fuzzy inference system on the launch zone (i.e., initial condition) of the PN guidance law. Tyan [12] presented a unified 3D PN guidance law, as well as its capture condition in terms of the closing speed components. However, the previous treatments on this subject are mainly based on some assumptions (i.e., target/missile speed ratio N > 1, constant speed, etc.), and the derived capture conditions are usually functions of time-to-go, navigation constant, speed ratio, or initial conditions. The capturability analysis of a post-launch missile guided by the PN guidance law has not
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C.-Y. Li, W.-X. Jing / Aerospace Science and Technology 12 (2008) 177–183
Nomenclature Nomenclature 3D DG LOS LOSR PN PPN TPN
r s t v ω
three-dimensional differential geometric line-of-sight LOS rate proportional navigation pure PN true PN
Subscript m r t 0 ω
Notation a e κ n N
acceleration vector unit vector with subscript curvature command with subscript unit normal vector with subscript target/missile speed ratio
missile along LOS vector target initial condition normal to LOS vector
Superscript
•
been fully addressed in the open literature, especially in terms of DG formulations. More specifically, there have not been many attempts on the application of the DG formulations to the missile guidance problems since Adler [13] extended the 2D PN guidance problem to three dimensions in terms of the geodesic and normal curvature. Bezick [14] introduced a nonlinear geometric guidance command using feedback linearization techniques. Leng [15] and Taur [16] developed a geometric guidance algorithm by introducing the relative heading error angle. The DG guidance commands developed by Chiou and Kuo [17–19] are based on the classical Frenet formulas [20], and were shown to be a generalization of PN guidance law [21,22]. Ariff [23] presented a novel DG guidance scheme using the involute information of the target’s trajectory. White [24] examined the direct intercept problem using DG formulations. This paper differs from the prior work in three main aspects. First, it focuses on the application of DG formulations to the capturability analysis of two classical PN guidance laws. Second, a unified, post-launch capture condition for the PN guidance law is developed and expressed in terms of geometric angle and relative motion parameters (i.e. closing speed and LOSR). Third, the proposed capture condition and the influence of the acceleration/deceleration of the missile to the interception performance of the PN guidance law are qualitatively studied in the simulation section.
derivative with respect to s derivative with respect to time
Fig. 1. Geometry of the engagement.
Taking derivative of Eq. (1) with respect to s, the following kinematics equations and its scalar components along the er and eω × er directions result: tm = N tt − r er − rω(eω × er )
In this section, the DG dynamics and kinematics formulations are briefly presented using the Frenet formulas [20]. In particular, the assumptions of point mass and constant speed are made in this section for both the missile and target in order to simplify this guidance problem. Referring to Fig. 1, we have (1)
(2)
r = (N tt − tm ) • er
(3)
rω = (N tt − tm ) • (eω × er )
(4)
Therefore, from Eq. (4), we have eω = (N tt − tm ) × er /(rω)
2. Differential geometric dynamics formulations
rm = rt − rer
LOS or position vector with subscript arc length along the missile’s trajectory unit tangent vector velocity vector with subscript LOSR
(5)
Furthermore, taking the derivative of Eq. (2) with respect to s, and applying the Frenet formulas, then we have the basic DG dynamics equations as follows: κm nm = N 2 κt nt − r er − 2r ωeω × er − rω eω × er − rωeω × er − rω2 eω × (eω × er )
(6)
The scalar components of Eq. (6) along the er and eω × er directions are
C.-Y. Li, W.-X. Jing / Aerospace Science and Technology 12 (2008) 177–183
r − rω2 = N 2 κt nt − κm nm • er
(7)
rω + 2r ω = (N κt nt − κm nm ) • (eω × er )
(8)
It should be pointed out that the vectors, tm and tt , are along the velocity directions, nm and nt are along the normal directions of the missile and target’s trajectories [20]. Moreover, according to the classical differential geometry theory [20], we have κm = |rm |,
nm = rm /|rm |
(9)
Hence κm nm = rm
d 2t d 2 rm 2 2 = v + am /vm = (am − tm v˙m )/vm m ds 2 ds 2
In the arc length system, the normal acceleration produced by the 3D PPN guidance law is [1,2,12,25] Dpn = (eω × tm )/|eω × tm |
(18)
Q = Aωvm
(19)
where A > 2 is the effective navigation ratio of the PN guidance law [1,2]. Therefore, substituting Eq. (18) into Eq. (17) 2 /|eω × tm |(eω × tm ) • er r = − Q/vm 2 + (at − amo ) • er /vm + rω2
(11)
2 + (at − amo ) • er /vm + rω2
(12)
r =
(13)
(15)
Also, in the same manner, we have 2 N 2 κt nt = at /vm
(16)
(22)
Consequently, we have r =
−Q (er × tm )2 − N (er × tm ) • (er × tt ) × tm |
2 |e rωvm ω
2 + rω2 + (at − amo ) • er /vm
(14)
where Q is the magnitude of the normal acceleration produced by the PN guidance law; Dpn is the direction of the normal acceleration produced by the PN guidance law; amo is the residual acceleration vector of the missile. Substituting Eq. (14) into Eq. (13), we have 2 κm nm = (QDpn + amo )/vm
−Q (er × tm ) × er • (tm − N tt ) × tm |
2 |e rωvm ω
2 + rω2 + (at − amo ) • er /vm
Moreover, the acceleration of the missile can be decomposed into two parts, formulated as follows: am = QDpn + amo
(21)
Furthermore, applying Eq. (5) to Eq. (21), results in
Recalling the constant speed assumption, we have 2 κm nm = am /vm
(20)
Hence 2 r = Q/vm /|eω × tm |(er × tm ) • eω
Therefore, 2 κm nm = rm = (am − tm v˙m )/vm
3.1. Capture analysis of PPN guidance law
(10)
Furthermore, in the time domain, we have rm =
179
(23)
Furthermore, according to Eq. (19), the right-hand side of Eq. (23) can be expressed in the time domain in the form: r =
−A[(er × tm )2 − N (er × tm ) • (er × tt )] rvm |eω × tm | 2 + (at − amo ) • er + rω(t)2 /vm
(24)
Note that the miss distance will keep decreasing if the closing speed keeps decreasing (i.e., r < 0). Therefore, in order to achieve a better performance, the right-hand side of Eq. (24) should be guaranteed to be negative throughout the guidance envelope, yielding
Substituting the preceding relations into Eq. (7), we have 2 r = at − (QDpn + amo ) • er /vm + rω2 (17)
(er × tm ) • (er × tt ) 0
Note that the last relation is the required simplified DG dynamics equation, which, compared with the conventional missile dynamics [1,2], indicates more clearly the relation between the miss distance and the geometric vectors, thus facilitating the capture analysis of the PN guidance law.
Referring to Fig. 2, Eq. (25) means that the velocities of the missile and target shall be placed on the opposite side of a plane determined by the LOS vector and the local horizontal axis (i.e., OI XI ), or one of them stays in this plane. Furthermore, according to the definition of the engagement, at and amo can be decomposed as follows:
3. Capture analysis of PN guidance law
at = aqt + apt + agt ,
A qualitatively study of the post-launch capturability of the PN guidance law is presented in this section. In particular, without loss of generality, the thrust, atmospheric force, and gravitation are considered over the post-launch engagement.
where aqt is the target’s acceleration vector caused by the atmospheric forces, apt and apm are the target and missile’s acceleration vectors produced by the thrust, respectively, agt and agm are the target and missile’s acceleration vectors caused by gravitation, respectively.
(25)
B = (at − amo ) • er + rω(t) < 0 2
amo = apm + agm
(26)
(27)
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In this way, Eq. (17) becomes 2 /|eω × er |(eω × er ) • er r = Q/vm 2 + (at − amo ) • er /vm + rω2
(35)
since
Fig. 2. Distribution of the velocity.
In the homing phase of the missile defense engagement, because the vertical range between the missile and the target is relatively small compared with the radius of the earth, thus the difference between agt and agm can be ignored. That is, agt ≈ agm . We further assume that there is no thrust for the target in this phase, i.e., apt = 0, and the target’s maneuver is caused mainly by the atmospheric force. Therefore, Eq. (26) becomes B = (aqt − apm ) • er + rω(t)2
(28)
Consequently, if the thrust is no longer available to the missile (i.e., apm = 0). Therefore, in this case, the following relation must be satisfied so as to render B < 0 aqt • er < −rω(t)2
(29)
Hence cos−1 −rω(t)2 /|aqt | < Ωm π
(30)
where Ωm is the angle between aqt and er . Obviously, Eq. (30) indicates that a better performance will result if the direction of the target’s maneuvering acceleration (i.e., aqt ) points towards the missile (i.e., along the opposite direction of LOS). Furthermore, in the case that the thrust is available to the missile during the guidance envelope, the following relation shall be guaranteed for a negative B apm • er > rω(t)2 + aqt • er
(31)
Consequently, we have |ηm | < cos−1 m rω(t)2 + χ /P
(32)
where χ is defined as the maximum permissible acceleration of the target (i.e., |aqt |max = χ), ηm is the angle between the LOS and the thrust, and 0 < |ηm | π/2. Therefore, if Eq. (25) and Eq. (30)/Eq. (32) are both satisfied, the right-hand side of Eq. (24) will maintain negative throughout the guidance envelope (i.e. r < 0), which means the miss distance keeps decreasing in this case, thus leading to a better performance. 3.2. Capture analysis of TPN guidance law In the arc length system, the normal acceleration produced by the 3D TPN guidance law is [1,2,12,25] D pn = (eω × er )/|eω × er |
(33)
Q = Arω
(34)
(eω × er ) • er = 0
(36)
Hence, Eq. (35) becomes 2 r = (at − amo ) • er + rω(t)2 /vm
(37)
Obviously, the same capture condition (i.e., Eqs. (30) and (32)) is applied for the TPN guidance law. Furthermore, if Eq. (32) becomes |ηm | < cos−1 m rω(t)2 + χ − ε /P (38) where ε < 0, and |ε| is a sufficiently small value. Then r < ε < 0 results. This means that the closing speed could be adjusted to a negative sufficiently small value in finite time, such that a better performance is achieved. It should also be pointed out that knowledge of the closing speed is necessary for the optimum solution of the navigation problem because the optimum value of the acceleration command to the autopilot is proportional to the missile’s closing speed. Therefore, Eq. (30) or Eq. (38) is the unified capture condition for the PN guidance law in the DG frame. It is apparent that the proposed capture condition is a necessary, postlaunch condition. Also, from the above analysis, it is clear that the existence of the thrust, or in other words, the acceleration/deceleration of the missile during the guidance envelope has a significant influence on the interception performance of the PN guidance law, which will be investigated in the simulation section. Moreover, it is well known that, theoretically, the maximum permissible target’s maneuverability for PN guidance law is one-third of the missile’s maximum allowable lateral acceleration [1,2], i.e., Λm . That is, χ Λm /3
(39)
Therefore, in the case of intercepting a target with low maneuverability, a more conservative capture condition can be derived in the form: |ηm | < cos−1 m rω(t)2 + (Λm /3) − ε /P (40) It should also be pointed out that the preceding capture analysis is another indicator of why the capture region of PPN guidance law is not as good as that of the TPN guidance law [1,3]. Moreover, as illustrated in Fig. 3, om is the center of mass of the missile, om xb is the directions of the longitudinal axis of the missile. For conventional aerodynamically controlled missiles, the direction of the thrust, i.e. Ip1 , usually coincides with the missile’s longitudinal axis. Therefore, ηm1 is equivalent to the angle between the LOS and the longitudinal axis (defined as the capture angle in this paper). Obviously, the preliminary explanation of Eq. (38) is that a better performance can be achieved provided that the longitudinal axis of the missile coincides with
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Fig. 3. Illustration of the guidance and thrust geometry.
the LOS. That is, a head-on interception scenario. Although, ηm1 cannot be controlled directly by the thrust in this case, the influence of the thrust on ηm1 and the interception performance is not negligible. Furthermore, for missiles whose thrust does not coincide with its longitudinal axis, a better performance can be guaranteed provided that the capture angle falls within the compact set determined by Eq. (38). 4. Simulation results To test the effectiveness of the proposed capture condition, and to study the influence of the missile’s acceleration/deceleration on the interception performance of the PN guidance law as well, simulation results are presented in this section for a missile defense engagement [22]. The initial conditions and constants for all of the following simulations are specified and listed below in units of meters, degrees, and seconds. Initial position (m) and velocity (m/s) of the missile: rm0 = [0; 0; 0],
The interception performance of the TPN guidance law and the PPN guidance law in both cases is compared and listed in Table 1, where MD means the miss distance. N-M denotes the case with a non-maneuvering target, in which the guidance angles (AOA and sideslip angle) of the target are maintained zero throughout the engagement. M denotes the case with a maneuvering target, in which the guidance angles of the target are time-varying. It should be pointed out that all performance results, except the gain column, are based on 200-run Monte Carlo sets, in which case, we assume that the LOSR is contaminated with 3σ uncertainty during the homing guidance envelope, and we further assume that the measurement error of the onboard seeker is 0.01 deg/s. As indicated in Table 1, regardless of the type of target, all the guidance schemes in Case II perform better than those in Case I, especially in the case of intercepting a maneuvering target, where the classical PN guidance law without thrust fails in capture because the maximum lateral acceleration of the target has already reached 14g in this case. However, the PN guidance law with a thrust during the guidance envelope can still guarantee an interception. Referring to Figs. 4 and 5, regardless of the type of guidance law, the capture angles in Case II (with thrust) are smaller than those in Case I (without thrust) at the end of the engagement, which means the longitudinal axis of the missile would stay close to the LOS in the case where the missile is accelerating towards the target. Otherwise, the capture angles will increase dramatically shortly after the fuel is burned out (21 s). In other Table 1 Comparison of interception performance MD (m)
vm0 = [0; 0; 0]
Initial position (m) and velocity (m/s) of the target: rt0 = [34000; 46000; 34000], vt0 = [−800; −1300; −800] Initial mass of the missile (Kg): 1000 Mass of the target (Kg): 500 Time constant of the guidance system (s): 0.3 Time constant of the seeker system (s): 0.2 Maximum permissible load factor (g): 20 Effective radius of active radar (Km): 40 Simulation step (s): 0.01 A generic skid-to-turn missile is introduced in the following characteristic cases, where the direction of the thrust coincides with the missile’s body axis. Case I: The thrust is not always available to the missile during the guidance envelope. The thrust (N):P = 65000, burn time (s): 21, and the impulse (s): 250. Case II: The thrust is available to the missile throughout the engagement. The thrust (N):P = 65000, burn time (s): 30, and the impulse (s): 250.
181
N-M
Time (s)
Gain
M
N-M
M
N-M
M
Case I
PPN TPN
3.21 2.94
23.77 21.55
24.47 24.53
24.78 24.73
5 5
8 9
Case II
PPN TPN
3.10 2.09
4.02 3.07
24.18 24.29
24.30 24.28
6 5
8 9
Fig. 4. Time history of the capture angles in the case of intercepting a non-maneuvering target.
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words, the longitudinal axis of the missile would leave the LOS when it is decelerating, thus leading to a worse performance. Moreover, the illustrations of the standard/benchmark capture angle, determined by the right-hand side of Eq. (38), indicate clearly that the proposed capture condition is a necessary condition, and the desired capture angle decreases gradually with the increase of the time, this is due to the fact that the magnitude of the LOSR increases with the decrease of the distance. As illustrated in Figs. 6 and 7, it is apparent that, regardless of the type of target or guidance law, the magnitude of the closing speed without the thrust tends to be smaller compared with the one with the thrust, which means the missile has a tendency of leaving the target when it decelerates [1]. This is another indicator of why the capture angle is relative bigger and the interception performance is worse in this case. It is worth stressing that noise has not been considered in this work. The fundamental effect of noise is to mask/hide the true value of the LOS. Noise can occur due to target effects
Fig. 7. Time history of the closing speed in the case of intercepting a maneuvering target.
or missile receiver effects. Target effects are scintillation and fading noise. In addition, the radome contributes a bias error due to the diffraction effects of the radome. 5. Conclusion
Fig. 5. Time history of the capture angles in the case of intercepting a maneuvering target.
In this paper, the capturability of the classical PN guidance law is presented using DG formulations. A unified, necessary post-launch capture condition is developed and expressed in terms of geometric angle and relative motion parameters, instead of the initial conditions. The simulation results demonstrate that the proposed capture condition is more crucial at the end of the engagement, and the acceleration/deceleration of the missile has a distinct influence on the performance of the PN guidance law. Moreover, a constant thrust in the homing guidance envelope works well and effectively in improving the interception performance of the PN guidance law. However, the derived capture conditions need further analysis, especially on its sufficient condition. Also, the results should be tested under uncertainties in the presence of a flight control system. Acknowledgements This work is supported in part by Shanghai Electro-Mechanical Engineering Institute research fund. Chaoyong Li is indebted to Dr. George M. Siouris, of Dayton, OH, USA, for his valuable suggestions on PN guidance problems, as well as for his careful review of this paper. The authors also appreciate an anonymous reviewer for his constructive comments and corrections. References
Fig. 6. Time history of the closing speed in the case of intercepting a non-maneuvering target.
[1] G.M. Siouris, Missile Guidance and Control Systems, Springer-Verlag, New York, 2004. [2] P. Zarchan, Tactical and Strategic Missile Guidance, Progress in Astronautics and Aeronautics, vol. 199, AIAA, 2002.
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[3] U.S. Shukla, P.R. Mahapatra, The proportional navigation dilemma—pure or true?, IEEE Trans. AES 26 (2) (1990) 382–392. [4] P.J. Yang, J.S. Chern, Solutions of true proportional navigation for maneuvering and nonmaneuvering targets, J. Guidance Control Dynam. 15 (1) (1992) 268–271. [5] D. Ghose, True proportional navigation with maneuvering target, IEEE Trans. AES 30 (1) (1994) 229–237. [6] S.H. Song, I.J. Ha, A Lyapunov-like approach to performance analysis of 3-dimensional pure PNG laws, IEEE Trans. AES 30 (1) (1994) 238–248. [7] J.H. Oh, I.J. Ha, Capturability of the 3-dimensional pure PNG law, IEEE Trans. AES 35 (2) (1999) 491–503. [8] S.N. Ghawghawe, D. Ghose, Pure proportional navigation against timevarying target maneuvers, IEEE Trans. AES 32 (4) (1996) 1336–1347. [9] C.D. Yang, C.C. Yang, Analytical solution of generalized threedimensional proportional navigation, J. Guidance Control Dynam. 19 (3) (1996) 721–724. [10] C.D. Yang, C.C. Yang, Analytical solution of 3D true proportional navigation, IEEE Trans. AES 32 (4) (1996) 1509–1522. [11] G.M. Dimirovski, S.M. Deskovski, Z.M. Gacovski, Classical and fuzzysystem guidance laws in homing missiles systems, in: Proc. IEEE Aerospace Conference, 2004. [12] F. Tyan, Unified approach to missile guidance law: A 3D extension, IEEE Trans. AES 41 (4) (2005) 1178–1199. [13] F.P. Adler, Missile guidance by three-dimensional proportional navigation, J. Appl. Phys. 27 (5) (1956) 500–507. [14] S. Bezick, I. Rusnak, Guidance of a homing missile via nonlinear geometric control method, J. Guidance Control Dynam. 18 (3) (1995) 441–448.
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[15] G. Leng, Guidance algorithm design: a nonlinear inverse approach, J. Guidance Control Dynam. 21 (5) (1998) 742–746. [16] D.R. Taur, Nonlinear guidance and navigation of a tactical missile with high heading error, AIAA-2002-4773. [17] C.Y. Kuo, Y.C. Chiou, Geometric analysis of missile guidance command, IEE Proc-Control Theory Appl. 147 (2) (2000) 205–211. [18] Y.C. Chiou, C.Y. Kuo, Geometric approach to three-dimensional missile guidance problem, J. Guidance Control Dynam. 21 (2) (1998) 335– 341. [19] C.Y. Kuo, D. Soetanto, Y.C. Chiou, Geometric analysis of flight control command for tactical missile guidance, IEEE Trans. Control System Technology 9 (2) (2001) 234–243. [20] D.J. Struik, Lectures on Classical Differential Geometry, Dover, New York, 1998. [21] C.Y. Li, W.X. Jing, H. Wang, Z.G. Qi, A novel approach to the 2D differential geometric guidance problem, Transactions of the Japan Society for Aeronautical and Space Sciences 50 (167) (2007) 34–40. [22] C.Y. Li, W.X. Jing, H. Wang, Z.G. Qi, Iterative solution to differential geometric guidance problem, Aircraft Engineering and Aerospace Technology 78 (5) (2006) 415–425. [23] O. Ariff, R. Zbikowski, A. Tsourdos, B.A. White, Differential geometric guidance based on the involute of the target’s trajectory, J. Guidance Control Dynam. 28 (5) (2005) 990–996. [24] B.A. White, O. Ariff, R. Zbikowski, A. Tsourdos, Direct intercept guidance using differential geometric concepts, AIAA -2005-5969. [25] A.K. Sarkar, P.K. Tiwari, S. Srinivasan, et al., Generalized PN guidance law for a practical pursuer evader engagement, AIAA-2003-5651.
Geometric approach to capture analysis of PN guidance law Chao-Yong Li ∗ , Wu-Xing Jing Department of Aerospace Engineering, Harbin Institute of Technology, P.O. Box 333, Harbin 150001, P.R. China Received 21 June 2006; received in revised form 16 April 2007; accepted 17 April 2007 Available online 16 May 2007
Abstract A novel approach to capture analysis of the proportional navigation (PN) guidance law is presented in this paper. The capturabilities of the pure PN guidance law and the true PN guidance law are qualitatively studied using the differential geometric formulations. A unified postlaunch capture condition for PN guidance law is developed and expressed in terms of geometric angle and relative motion parameters instead of the initial conditions. Simulation results indicate that the proposed capture condition is more crucial at the end of the engagement. Also, the acceleration/deceleration of the missile has a distinct influence on the interception performance of the PN guidance law, and a constant thrust works viable and effectively in improving the performance of the PN guidance law. © 2007 Elsevier Masson SAS. All rights reserved. Keywords: Missile guidance; Differential geometry; Proportional navigation; Capture condition
1. Introduction Because of its inherent simplicity and ease of implementation, PN guidance law and its variants are the most widely known and used guidance schemes in the practical missile interception engagements. In attempting to solve the PN equations, two principal PN guidance laws are defined due to the directions of their accelerations. One set of definitions leads to a class of laws consisting of PPN guidance law and its variants, whose commanded accelerations are referenced relative to the missile’s velocity,1 the other class is TPN guidance law and its variants, whose commanded accelerations are referenced relative to the LOS. Moreover, it is well know that notwithstanding the relative difficulties in solving the PPN guidance problems, PPN guidance law is the most “natural” guidance law, this conclusion is drawn based on the facts that the velocity-referenced guidance laws are reasonable in practical implementation [1–3]. Moreover, in view of the importance of the PN guidance law in the missile guidance and space applications, considerable studies have been devoted to its capturability analysis * Corresponding author,
E-mail address: [email protected] (C.-Y. Li). 1 Throughout this paper, the word velocity will only be used to designate a
vector quantity; the corresponding scalar will be denoted as speed. 1270-9638/$ – see front matter © 2007 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ast.2007.04.007
in various scenarios. Shukla [3] proposed a comparative study between the TPN and PPN guidance laws, as well as their capture conditions. Yang and Chern [4] examined the relation between the navigation constant and the capture condition of the TPN guidance law. Ghose [5] developed a necessary and sufficient capture condition for TPN guidance law in the case of intercepting an intelligently maneuvering target. Song [6] and Ha [7] introduced a Lyapunov-like approach to the capturability analysis of 3D PN guidance law. Ghawghawe [8] studied the capturability of the PPN guidance law against a target executing bounded piecewise continuous time-varying maneuvers. Yang [9] and Yang [10] analyzed the influence of the navigation constant, target’s maneuver, and the initial conditions on the capture condition of 3D PN guidance law. Dimirovski [11] investigated the influence of the application of the fuzzy inference system on the launch zone (i.e., initial condition) of the PN guidance law. Tyan [12] presented a unified 3D PN guidance law, as well as its capture condition in terms of the closing speed components. However, the previous treatments on this subject are mainly based on some assumptions (i.e., target/missile speed ratio N > 1, constant speed, etc.), and the derived capture conditions are usually functions of time-to-go, navigation constant, speed ratio, or initial conditions. The capturability analysis of a post-launch missile guided by the PN guidance law has not
178
C.-Y. Li, W.-X. Jing / Aerospace Science and Technology 12 (2008) 177–183
Nomenclature Nomenclature 3D DG LOS LOSR PN PPN TPN
r s t v ω
three-dimensional differential geometric line-of-sight LOS rate proportional navigation pure PN true PN
Subscript m r t 0 ω
Notation a e κ n N
acceleration vector unit vector with subscript curvature command with subscript unit normal vector with subscript target/missile speed ratio
missile along LOS vector target initial condition normal to LOS vector
Superscript
•
been fully addressed in the open literature, especially in terms of DG formulations. More specifically, there have not been many attempts on the application of the DG formulations to the missile guidance problems since Adler [13] extended the 2D PN guidance problem to three dimensions in terms of the geodesic and normal curvature. Bezick [14] introduced a nonlinear geometric guidance command using feedback linearization techniques. Leng [15] and Taur [16] developed a geometric guidance algorithm by introducing the relative heading error angle. The DG guidance commands developed by Chiou and Kuo [17–19] are based on the classical Frenet formulas [20], and were shown to be a generalization of PN guidance law [21,22]. Ariff [23] presented a novel DG guidance scheme using the involute information of the target’s trajectory. White [24] examined the direct intercept problem using DG formulations. This paper differs from the prior work in three main aspects. First, it focuses on the application of DG formulations to the capturability analysis of two classical PN guidance laws. Second, a unified, post-launch capture condition for the PN guidance law is developed and expressed in terms of geometric angle and relative motion parameters (i.e. closing speed and LOSR). Third, the proposed capture condition and the influence of the acceleration/deceleration of the missile to the interception performance of the PN guidance law are qualitatively studied in the simulation section.
derivative with respect to s derivative with respect to time
Fig. 1. Geometry of the engagement.
Taking derivative of Eq. (1) with respect to s, the following kinematics equations and its scalar components along the er and eω × er directions result: tm = N tt − r er − rω(eω × er )
In this section, the DG dynamics and kinematics formulations are briefly presented using the Frenet formulas [20]. In particular, the assumptions of point mass and constant speed are made in this section for both the missile and target in order to simplify this guidance problem. Referring to Fig. 1, we have (1)
(2)
r = (N tt − tm ) • er
(3)
rω = (N tt − tm ) • (eω × er )
(4)
Therefore, from Eq. (4), we have eω = (N tt − tm ) × er /(rω)
2. Differential geometric dynamics formulations
rm = rt − rer
LOS or position vector with subscript arc length along the missile’s trajectory unit tangent vector velocity vector with subscript LOSR
(5)
Furthermore, taking the derivative of Eq. (2) with respect to s, and applying the Frenet formulas, then we have the basic DG dynamics equations as follows: κm nm = N 2 κt nt − r er − 2r ωeω × er − rω eω × er − rωeω × er − rω2 eω × (eω × er )
(6)
The scalar components of Eq. (6) along the er and eω × er directions are
C.-Y. Li, W.-X. Jing / Aerospace Science and Technology 12 (2008) 177–183
r − rω2 = N 2 κt nt − κm nm • er
(7)
rω + 2r ω = (N κt nt − κm nm ) • (eω × er )
(8)
It should be pointed out that the vectors, tm and tt , are along the velocity directions, nm and nt are along the normal directions of the missile and target’s trajectories [20]. Moreover, according to the classical differential geometry theory [20], we have κm = |rm |,
nm = rm /|rm |
(9)
Hence κm nm = rm
d 2t d 2 rm 2 2 = v + am /vm = (am − tm v˙m )/vm m ds 2 ds 2
In the arc length system, the normal acceleration produced by the 3D PPN guidance law is [1,2,12,25] Dpn = (eω × tm )/|eω × tm |
(18)
Q = Aωvm
(19)
where A > 2 is the effective navigation ratio of the PN guidance law [1,2]. Therefore, substituting Eq. (18) into Eq. (17) 2 /|eω × tm |(eω × tm ) • er r = − Q/vm 2 + (at − amo ) • er /vm + rω2
(11)
2 + (at − amo ) • er /vm + rω2
(12)
r =
(13)
(15)
Also, in the same manner, we have 2 N 2 κt nt = at /vm
(16)
(22)
Consequently, we have r =
−Q (er × tm )2 − N (er × tm ) • (er × tt ) × tm |
2 |e rωvm ω
2 + rω2 + (at − amo ) • er /vm
(14)
where Q is the magnitude of the normal acceleration produced by the PN guidance law; Dpn is the direction of the normal acceleration produced by the PN guidance law; amo is the residual acceleration vector of the missile. Substituting Eq. (14) into Eq. (13), we have 2 κm nm = (QDpn + amo )/vm
−Q (er × tm ) × er • (tm − N tt ) × tm |
2 |e rωvm ω
2 + rω2 + (at − amo ) • er /vm
Moreover, the acceleration of the missile can be decomposed into two parts, formulated as follows: am = QDpn + amo
(21)
Furthermore, applying Eq. (5) to Eq. (21), results in
Recalling the constant speed assumption, we have 2 κm nm = am /vm
(20)
Hence 2 r = Q/vm /|eω × tm |(er × tm ) • eω
Therefore, 2 κm nm = rm = (am − tm v˙m )/vm
3.1. Capture analysis of PPN guidance law
(10)
Furthermore, in the time domain, we have rm =
179
(23)
Furthermore, according to Eq. (19), the right-hand side of Eq. (23) can be expressed in the time domain in the form: r =
−A[(er × tm )2 − N (er × tm ) • (er × tt )] rvm |eω × tm | 2 + (at − amo ) • er + rω(t)2 /vm
(24)
Note that the miss distance will keep decreasing if the closing speed keeps decreasing (i.e., r < 0). Therefore, in order to achieve a better performance, the right-hand side of Eq. (24) should be guaranteed to be negative throughout the guidance envelope, yielding
Substituting the preceding relations into Eq. (7), we have 2 r = at − (QDpn + amo ) • er /vm + rω2 (17)
(er × tm ) • (er × tt ) 0
Note that the last relation is the required simplified DG dynamics equation, which, compared with the conventional missile dynamics [1,2], indicates more clearly the relation between the miss distance and the geometric vectors, thus facilitating the capture analysis of the PN guidance law.
Referring to Fig. 2, Eq. (25) means that the velocities of the missile and target shall be placed on the opposite side of a plane determined by the LOS vector and the local horizontal axis (i.e., OI XI ), or one of them stays in this plane. Furthermore, according to the definition of the engagement, at and amo can be decomposed as follows:
3. Capture analysis of PN guidance law
at = aqt + apt + agt ,
A qualitatively study of the post-launch capturability of the PN guidance law is presented in this section. In particular, without loss of generality, the thrust, atmospheric force, and gravitation are considered over the post-launch engagement.
where aqt is the target’s acceleration vector caused by the atmospheric forces, apt and apm are the target and missile’s acceleration vectors produced by the thrust, respectively, agt and agm are the target and missile’s acceleration vectors caused by gravitation, respectively.
(25)
B = (at − amo ) • er + rω(t) < 0 2
amo = apm + agm
(26)
(27)
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In this way, Eq. (17) becomes 2 /|eω × er |(eω × er ) • er r = Q/vm 2 + (at − amo ) • er /vm + rω2
(35)
since
Fig. 2. Distribution of the velocity.
In the homing phase of the missile defense engagement, because the vertical range between the missile and the target is relatively small compared with the radius of the earth, thus the difference between agt and agm can be ignored. That is, agt ≈ agm . We further assume that there is no thrust for the target in this phase, i.e., apt = 0, and the target’s maneuver is caused mainly by the atmospheric force. Therefore, Eq. (26) becomes B = (aqt − apm ) • er + rω(t)2
(28)
Consequently, if the thrust is no longer available to the missile (i.e., apm = 0). Therefore, in this case, the following relation must be satisfied so as to render B < 0 aqt • er < −rω(t)2
(29)
Hence cos−1 −rω(t)2 /|aqt | < Ωm π
(30)
where Ωm is the angle between aqt and er . Obviously, Eq. (30) indicates that a better performance will result if the direction of the target’s maneuvering acceleration (i.e., aqt ) points towards the missile (i.e., along the opposite direction of LOS). Furthermore, in the case that the thrust is available to the missile during the guidance envelope, the following relation shall be guaranteed for a negative B apm • er > rω(t)2 + aqt • er
(31)
Consequently, we have |ηm | < cos−1 m rω(t)2 + χ /P
(32)
where χ is defined as the maximum permissible acceleration of the target (i.e., |aqt |max = χ), ηm is the angle between the LOS and the thrust, and 0 < |ηm | π/2. Therefore, if Eq. (25) and Eq. (30)/Eq. (32) are both satisfied, the right-hand side of Eq. (24) will maintain negative throughout the guidance envelope (i.e. r < 0), which means the miss distance keeps decreasing in this case, thus leading to a better performance. 3.2. Capture analysis of TPN guidance law In the arc length system, the normal acceleration produced by the 3D TPN guidance law is [1,2,12,25] D pn = (eω × er )/|eω × er |
(33)
Q = Arω
(34)
(eω × er ) • er = 0
(36)
Hence, Eq. (35) becomes 2 r = (at − amo ) • er + rω(t)2 /vm
(37)
Obviously, the same capture condition (i.e., Eqs. (30) and (32)) is applied for the TPN guidance law. Furthermore, if Eq. (32) becomes |ηm | < cos−1 m rω(t)2 + χ − ε /P (38) where ε < 0, and |ε| is a sufficiently small value. Then r < ε < 0 results. This means that the closing speed could be adjusted to a negative sufficiently small value in finite time, such that a better performance is achieved. It should also be pointed out that knowledge of the closing speed is necessary for the optimum solution of the navigation problem because the optimum value of the acceleration command to the autopilot is proportional to the missile’s closing speed. Therefore, Eq. (30) or Eq. (38) is the unified capture condition for the PN guidance law in the DG frame. It is apparent that the proposed capture condition is a necessary, postlaunch condition. Also, from the above analysis, it is clear that the existence of the thrust, or in other words, the acceleration/deceleration of the missile during the guidance envelope has a significant influence on the interception performance of the PN guidance law, which will be investigated in the simulation section. Moreover, it is well known that, theoretically, the maximum permissible target’s maneuverability for PN guidance law is one-third of the missile’s maximum allowable lateral acceleration [1,2], i.e., Λm . That is, χ Λm /3
(39)
Therefore, in the case of intercepting a target with low maneuverability, a more conservative capture condition can be derived in the form: |ηm | < cos−1 m rω(t)2 + (Λm /3) − ε /P (40) It should also be pointed out that the preceding capture analysis is another indicator of why the capture region of PPN guidance law is not as good as that of the TPN guidance law [1,3]. Moreover, as illustrated in Fig. 3, om is the center of mass of the missile, om xb is the directions of the longitudinal axis of the missile. For conventional aerodynamically controlled missiles, the direction of the thrust, i.e. Ip1 , usually coincides with the missile’s longitudinal axis. Therefore, ηm1 is equivalent to the angle between the LOS and the longitudinal axis (defined as the capture angle in this paper). Obviously, the preliminary explanation of Eq. (38) is that a better performance can be achieved provided that the longitudinal axis of the missile coincides with
C.-Y. Li, W.-X. Jing / Aerospace Science and Technology 12 (2008) 177–183
Fig. 3. Illustration of the guidance and thrust geometry.
the LOS. That is, a head-on interception scenario. Although, ηm1 cannot be controlled directly by the thrust in this case, the influence of the thrust on ηm1 and the interception performance is not negligible. Furthermore, for missiles whose thrust does not coincide with its longitudinal axis, a better performance can be guaranteed provided that the capture angle falls within the compact set determined by Eq. (38). 4. Simulation results To test the effectiveness of the proposed capture condition, and to study the influence of the missile’s acceleration/deceleration on the interception performance of the PN guidance law as well, simulation results are presented in this section for a missile defense engagement [22]. The initial conditions and constants for all of the following simulations are specified and listed below in units of meters, degrees, and seconds. Initial position (m) and velocity (m/s) of the missile: rm0 = [0; 0; 0],
The interception performance of the TPN guidance law and the PPN guidance law in both cases is compared and listed in Table 1, where MD means the miss distance. N-M denotes the case with a non-maneuvering target, in which the guidance angles (AOA and sideslip angle) of the target are maintained zero throughout the engagement. M denotes the case with a maneuvering target, in which the guidance angles of the target are time-varying. It should be pointed out that all performance results, except the gain column, are based on 200-run Monte Carlo sets, in which case, we assume that the LOSR is contaminated with 3σ uncertainty during the homing guidance envelope, and we further assume that the measurement error of the onboard seeker is 0.01 deg/s. As indicated in Table 1, regardless of the type of target, all the guidance schemes in Case II perform better than those in Case I, especially in the case of intercepting a maneuvering target, where the classical PN guidance law without thrust fails in capture because the maximum lateral acceleration of the target has already reached 14g in this case. However, the PN guidance law with a thrust during the guidance envelope can still guarantee an interception. Referring to Figs. 4 and 5, regardless of the type of guidance law, the capture angles in Case II (with thrust) are smaller than those in Case I (without thrust) at the end of the engagement, which means the longitudinal axis of the missile would stay close to the LOS in the case where the missile is accelerating towards the target. Otherwise, the capture angles will increase dramatically shortly after the fuel is burned out (21 s). In other Table 1 Comparison of interception performance MD (m)
vm0 = [0; 0; 0]
Initial position (m) and velocity (m/s) of the target: rt0 = [34000; 46000; 34000], vt0 = [−800; −1300; −800] Initial mass of the missile (Kg): 1000 Mass of the target (Kg): 500 Time constant of the guidance system (s): 0.3 Time constant of the seeker system (s): 0.2 Maximum permissible load factor (g): 20 Effective radius of active radar (Km): 40 Simulation step (s): 0.01 A generic skid-to-turn missile is introduced in the following characteristic cases, where the direction of the thrust coincides with the missile’s body axis. Case I: The thrust is not always available to the missile during the guidance envelope. The thrust (N):P = 65000, burn time (s): 21, and the impulse (s): 250. Case II: The thrust is available to the missile throughout the engagement. The thrust (N):P = 65000, burn time (s): 30, and the impulse (s): 250.
181
N-M
Time (s)
Gain
M
N-M
M
N-M
M
Case I
PPN TPN
3.21 2.94
23.77 21.55
24.47 24.53
24.78 24.73
5 5
8 9
Case II
PPN TPN
3.10 2.09
4.02 3.07
24.18 24.29
24.30 24.28
6 5
8 9
Fig. 4. Time history of the capture angles in the case of intercepting a non-maneuvering target.
182
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words, the longitudinal axis of the missile would leave the LOS when it is decelerating, thus leading to a worse performance. Moreover, the illustrations of the standard/benchmark capture angle, determined by the right-hand side of Eq. (38), indicate clearly that the proposed capture condition is a necessary condition, and the desired capture angle decreases gradually with the increase of the time, this is due to the fact that the magnitude of the LOSR increases with the decrease of the distance. As illustrated in Figs. 6 and 7, it is apparent that, regardless of the type of target or guidance law, the magnitude of the closing speed without the thrust tends to be smaller compared with the one with the thrust, which means the missile has a tendency of leaving the target when it decelerates [1]. This is another indicator of why the capture angle is relative bigger and the interception performance is worse in this case. It is worth stressing that noise has not been considered in this work. The fundamental effect of noise is to mask/hide the true value of the LOS. Noise can occur due to target effects
Fig. 7. Time history of the closing speed in the case of intercepting a maneuvering target.
or missile receiver effects. Target effects are scintillation and fading noise. In addition, the radome contributes a bias error due to the diffraction effects of the radome. 5. Conclusion
Fig. 5. Time history of the capture angles in the case of intercepting a maneuvering target.
In this paper, the capturability of the classical PN guidance law is presented using DG formulations. A unified, necessary post-launch capture condition is developed and expressed in terms of geometric angle and relative motion parameters, instead of the initial conditions. The simulation results demonstrate that the proposed capture condition is more crucial at the end of the engagement, and the acceleration/deceleration of the missile has a distinct influence on the performance of the PN guidance law. Moreover, a constant thrust in the homing guidance envelope works well and effectively in improving the interception performance of the PN guidance law. However, the derived capture conditions need further analysis, especially on its sufficient condition. Also, the results should be tested under uncertainties in the presence of a flight control system. Acknowledgements This work is supported in part by Shanghai Electro-Mechanical Engineering Institute research fund. Chaoyong Li is indebted to Dr. George M. Siouris, of Dayton, OH, USA, for his valuable suggestions on PN guidance problems, as well as for his careful review of this paper. The authors also appreciate an anonymous reviewer for his constructive comments and corrections. References
Fig. 6. Time history of the closing speed in the case of intercepting a non-maneuvering target.
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